Three vertices of a tetrahedron are ( 0,0,0 )( 6, - 5, - 1 ) and ( - 4,1,3 ). If the centroid of the tetrahedron be ( 1, - 2,5 ) then the fourth vertex is :

If the fourth vertex is ( , , ) then centroid = ( 0 + 6 - 4 + 4,0 - 5 + 1 + 4,0 - 1 + 3 + 4 ) = ( 1, - 2,5 ).

three vertices of a tetrahedron are 000 6 5 1 and 413 if

Three vertices of a tetrahedron are $$\left( {0,\,0,\,0} \right)\left( {6,\, - 5,\, - 1} \right)$$ and $$\left( { - 4,\,1,\,3} \right).$$ If the centroid of the tetrahedron be $$\left( {1,\, - 2,\,5} \right)$$ then the fourth vertex is :

A. $$\left( {2,\, - 4,\,18} \right)$$

B. $$\left( {2,\, - 4,\, - 18} \right)$$

C. $$\left( {\frac{3}{4},\,\frac{{ - 3}}{2},\,\frac{7}{4}} \right)$$

D. none of these

Answer : $$\left( {2,\, - 4,\,18} \right)$$

Solution :
If the fourth vertex is $$\left( {\alpha ,\,\beta ,\,\gamma } \right)$$ then centroid $$ = \left( {\frac{{0 + 6 - 4 + \alpha }}{4},\,\frac{{0 - 5 + 1 + \beta }}{4},\,\frac{{0 - 1 + 3 + \gamma }}{4}} \right) = \left( {1,\, - 2,\,5} \right).$$

Releted MCQ Question on
Geometry >> 3D Geometry and Vectors

Releted Question 1

The scalar $$\vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right)$$ equals :

A. $$0$$

B. $$\left[ {\vec A\,\vec B\,\vec C} \right] + \left[ {\vec B\,\vec C\,\vec A} \right]$$

C. $$\left[ {\vec A\,\vec B\,\vec C} \right]$$

D. None of these

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Releted Question 2

For non-zero vectors $$\vec a,\,\vec b,\,\vec c,\,\left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right|$$ holds if and only if -

A. $$\vec a.\vec b = 0,\,\,\,\vec b.\vec c = 0$$

B. $$\vec b.\vec c = 0,\,\,\,\vec c.\vec a = 0$$

C. $$\vec c.\vec a = 0,\,\,\,\vec a.\vec b = 0$$

D. $$\vec a.\vec b = \vec b.\vec c = \vec c.\vec a = 0$$

View Answer

Releted Question 3

The volume of the parallelepiped whose sides are given by $$\overrightarrow {OA} = 2i - 2j,\,\,\overrightarrow {OB} = i + j - k,\,\,\overrightarrow {OC} = 3i - k,$$ is :

A. $$\frac{4}{{13}}$$

B. $$4$$

C. $$\frac{2}{7}$$

D. none of these

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Releted Question 4

The points with position vectors $$60i + 3j,\,\,40i - 8j,\,\,ai - 52j$$ are collinear if :

A. $$a = - 40$$

B. $$a = 40$$

C. $$a = 20$$

D. none of these

View Answer

Three vertices of a tetrahedron are $$\left( {0,\,0,\,0} \right)\left( {6,\, - 5,\, - 1} \right)$$ and $$\left( { - 4,\,1,\,3} \right).$$ If the centroid of the tetrahedron be $$\left( {1,\, - 2,\,5} \right)$$ then the fourth vertex is :

Releted MCQ Question on
Geometry >> 3D Geometry and Vectors

The scalar $$\vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right)$$ equals :

For non-zero vectors $$\vec a,\,\vec b,\,\vec c,\,\left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right|$$ holds if and only if -

The volume of the parallelepiped whose sides are given by $$\overrightarrow {OA} = 2i - 2j,\,\,\overrightarrow {OB} = i + j - k,\,\,\overrightarrow {OC} = 3i - k,$$ is :

The points with position vectors $$60i + 3j,\,\,40i - 8j,\,\,ai - 52j$$ are collinear if :

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Three vertices of a tetrahedron are $$\left( {0,\,0,\,0} \right)\left( {6,\, - 5,\, - 1} \right)$$ and $$\left( { - 4,\,1,\,3} \right).$$ If the centroid of the tetrahedron be $$\left( {1,\, - 2,\,5} \right)$$ then the fourth vertex is :

Releted MCQ Question on Geometry >> 3D Geometry and Vectors

The scalar $$\vec A.\left( {\vec B + \vec C} \right) \times \left( {\vec A + \vec B + \vec C} \right)$$ equals :

For non-zero vectors $$\vec a,\,\vec b,\,\vec c,\,\left| {\left( {\vec a \times \vec b} \right).\vec c} \right| = \left| {\vec a} \right|\left| {\vec b} \right|\left| {\vec c} \right|$$ holds if and only if -

The volume of the parallelepiped whose sides are given by $$\overrightarrow {OA} = 2i - 2j,\,\,\overrightarrow {OB} = i + j - k,\,\,\overrightarrow {OC} = 3i - k,$$ is :

The points with position vectors $$60i + 3j,\,\,40i - 8j,\,\,ai - 52j$$ are collinear if :

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Practice More MCQ Question on Maths Section

Releted MCQ Question on
Geometry >> 3D Geometry and Vectors

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3D Geometry and Vectors